Electronic – LED: should I smooth the current with a capacitor

please help me

I have some concerns about the current limit circuit (based around the BC337 / R2) and its ability to work with PWM - every time the pulse goes high from U3 (initiating a current flow through the MOSFET), the voltage developed across R2 will activate the BC337 and clamp the drive voltage to the MOSFET. This means the MOSFET will get quite warm because it is operating as a current limit device.

Added to this you are trying to control the LED current via PWM BUT the BC337 circuit is also trying to limit the current. It doesn't look like it should work as you expect.

When you say the LED starts blinking, what is your PWM frequency?

Given the circuit you have I also have concerns about possible high voltages from the LED string being capacitively coupled (through the MOSFET) back to an IO pin on your MCU. Maybe protection diodes would help.

Short answer: Current stops when the capacitor gets charged up to the battery voltage.

When current flows through the circuit, the bulb lights up. In this case you can consider the bulb as a 'current detector'. Current in this case is flow of charge, and the charge carriers are electrons. They are pushed around because the battery pushes them with an electric field.

Now look at the capacitor symbol. It indicates that there is a gap between the plates. Electrons normally don't jump this gap. But you know there is current because the bulb initially turns on. So what is happening? Electrons are effectively being accumulated at one of the plates of the capacitor, while the opposite plate has electrons being removed, "charging it". But they can't accumulate indefinitely, because the more they accumulate, the more the repel each other. It basically takes energy/work to do this, and you can consider that the capacitor is storing this energy, and can effectively be released later on. As the capacitor gets charged, the voltage accross it augments, until the battery cannot push more electrons. At this point the capacitor voltage has equalized the battery voltage. No more electrons flow, the bulb finally turns off.

To release the energy stored in the capacitor, remove the battery from the circuit and connect the wires together. You should see the same effect (bulb turning on, then dimming until off), because the battery no longer keeps those electrons pushed and they return to neutrality via the bulb.